Polynomial Functions, Zeros, Factors and Intercepts (1)

Tutorial and problems with detailed solutions on finding polynomial functions given their zeros and/or graphs and other information. Problems related to polynomials with real coefficients and complex solutions are also included.

Review

A) Let p(x) be a polynomial function with real coefficients. If p(s) = 0
  1. s is a zero for the polynomial function p(x).
  2. s is a solution to the equation p(x) = 0
  3. (x - s) is a factor of p(x).
  4. The point (s , 0) is an x intercept of the graph of p(x).

B) In what follows the imaginary unit i is defined as
i = √(-1)
Let p(x) be a polynomial function with real coefficients. If a + ib is an complex zero of p(x), the conjugate a - bi is also a zero of p(x).

Example

let p(x) = x3 - 2 x2 + 2 x - 4
  1. p(2) = 23 - 2*22 + 2*2 - 4
    = 8 - 8 + 4 - 4
    = 0
    2 is a zero of p(x).
  2. x = 2 is a solution of p(x) = 0
  3. p(x) can be written in factored form as
    p(x) =(x - 2) (x2 + 2)
  4. The graph of p(x) below shows an x intercept at x = 2.

Graph of p(x), example above.

Problems with Solutions

Problem 1

The graph below is that of a polynomial function p(x) with real coefficients. The degree of p(x) is 3 and the zeros are assumed to be integers. Find p(x).
Graph of p(x), problem 1.
Solution to Problem 1
The graph has 2 x intercepts: -1 and 2. The x intercept at -1 is of multiplicity 2. p(x) can be written as follows
p(x) = a(x + 1)
2(x - 2) , a is any real constant not equal to zero.
To find a we need to use more information in the graph. The y intercept is at (0 , -2), which means that p(0) = -2
a(0 + 1)
2(0 - 2) = -2
Solve the above equation for a to obtain
a = 1
p(x) is given by
p(x) = (x + 1)
2(x - 2)

Problem 2

A polynomial function p(x) with real coefficients and of degree 5 has the zeros: -1, 2(with multiplicity 2) , 0 and 1. p(3) = -12. Find p(x).
Solution to Problem 2
p(x) can be written as follows
p(x) = a x(x + 1)(x - 2)
2(x - 1) , a is any real constant not equal to zero.
p(3) = -12 gives the following equation in a.
a(3)(3 + 1)(3 - 2)
2(3 - 1) = -12
Solve the above equation for a to obtain
a = -1/2
p(x) is given by
p(x) = -0.5 x(x + 1)(x - 2)
2(x - 1)
The graph of p(x) is shown below.
Check the intercepts and the point (3 , -12) on the graph of p(x) found above.

Problem 3

2 + i is a zero of polynomial p(x) given below, find all the other zeros.
p(x) = x4 - 2 x3 - 6 x2 + 22 x - 15

Solution to Problem 3
The zero 2 + i is a complex number and p(x) has real coefficients. It follows that the conjugate 2 - i is also a zero of p(x). p(x) may be written in factored form as follows
p(x) = (x - (2 + i)) (x - (2 - i)) q(x)
Let us expand the term (x - (2 + i)) (x - (2 - i)) in p(x)
(x - (2 + i)) (x - (2 - i)) = x
2 -(2 + i)x -(2 - i)x + (2+i)(2-i)
= x
2 - 4 x + 5
q(x) can be found by dividing p(x) by x2 - 4 x + 5.
(x
4 - 2 x3 - 6 x2 + 22 x - 15) / (x2 - 4 x + 5)
= x
2 + 2 x - 3
We now write p(x) in factored form
p(x) = (x - (2 + i)) (x - (2 - i)) (x
2 + 2 x - 3)
The remaining 2 zeros of p(x) are the solutions to the quadratic equation.
x
2 + 2 x - 3 = 0
Factor the above quadratic equation and solve.
(x - 1) (x + 3) = 0
solutions
x = 1
x = -3
p(x) has the following zeros.
2 + i , 2 - i, -3 and 1.

Problem 4

-3 - i is a zero of polynomial p(x) given below, find all the other zeros.
p(x) = x4 + 6 x3 + 11 x2 + 6 x + 10

Solution to Problem 4
The zero -3 - i is a complex number and p(x) has real coefficients. Hence the conjugate -3 + i is also a zero of p(x). The factored form of p(x) is as follows
p(x) = (x - (-3 - i)) (x - (-3 + i)) q(x)
Let us expand the term (x - (-3 - i)) (x - (-3 + i)) in p(x)
(x - (-3 - i)) (x - (-3 + i)) = x
2 + 6 x + 10
q(x) can be found by dividing p(x) by x2 + 6 x + 10.
(x
4 + 6 x3 + 11 x2 + 6 x + 10) / (x2 + 6 x + 10)
= x
2 + 1
We now write p(x) in factored form
p(x) = (x - (2 + i)) (x - (2 - i)) (x
2 + 1)
The remaining 2 zeros of p(x) are the solutions to the quadratic equation.
x
2 + 1 = 0
Factor the above quadratic equation and solve.
x
2 + 1 = (x - i)(x + i)
solutions
x = i
x = - i
p(x) has the following zeros.
- 3 - i , - 3 + i, i and - i.

More References and Links to Polynomial Functions

Factor Polynomials.
Polynomial Functions.
Multiplicity of Zeros and Graphs Polynomials.
Find Zeros of Polynomial Functions - Problems
Graphs of Polynomial Functions - Self Test.
Step by Step Solver to Find a Polynomial Given its Zeros and a Point.
Step by Step Solver to Factor a Cubic Polynomial Given one of its Zeros.
Find a Cubic Polynomial Passing Through Four Points.